# Parallel Law for Extremal Length

## Theorem

Let $X$ be a Riemann surface.

Let $\Gamma_1, \Gamma_2$ be families of rectifiable curves (or, more generally, families of disjoint unions of rectifiable curves) on $X$.

Let $\Gamma_1$ and $\Gamma_2$ be disjoint, in the sense that:

there exist disjoint Borel subsets $A_1, A_2 \subseteq X$ such that:
for any $\gamma_1 \in \Gamma_1$ and $\gamma_2 \in \Gamma_2$, we have $\gamma_1 \subseteq A_1$ and $\gamma_2 \subseteq A_2$.

Let $\Gamma$ be a third curve family, with the property that every element $\Gamma_1$ and every element of $\Gamma_2$ contains some element of $\Gamma$.

Then the extremal length of $\Gamma$ satisfies:

$\dfrac 1 {\lambda \left({\Gamma}\right)} \ge \dfrac 1 {\lambda \left({\Gamma_1}\right)} + \dfrac 1 {\lambda \left({\Gamma_2}\right)}$

## Proof

The assumption means that every element of $\Gamma_1 \cup \Gamma_2$ contains some element of $\Gamma$.

Hence:

 $\ds \frac 1 {\lambda \left({\Gamma}\right)}$ $\ge$ $\ds \frac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)}$ Comparison Principle for Extremal Length $\ds$ $=$ $\ds \frac 1 {\lambda \left({\Gamma_1}\right)} + \frac 1 {\lambda \left({\Gamma_2}\right)}$ Extremal Length of Union

$\blacksquare$